Quartic Fields Ramifying Above {23, 47}

Discriminant class: −23

∞ 23 47  a_{1} a_{2} a_{3}  Over p = 23  Over p = 47  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 41 43 53 59 61
1. 1_{k} 3_{k} 2_{−}  -2 13 -12 1^{4} 3_{k} 23∗ 23 1^{2} 1_{i} 47∗ 47∗ 22 22 2 4 2 22 4 2
D_{4} -1045 1^{2} 1_{i} 47∗ 47∗ 22 22 4 22 2 4 22 4

Discriminant class: 1

∞ 23 47 a_{1} a_{2} a_{3} Over p = 23  Over p = 47  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d s f^{e} c_{w} d  s 29 31 37 41 43 53 59 61
2. 2_{−} 2_{+} 2_{−} 0 35 0 2^{2} 2_{+} 1 V 1^{2} 1_{i} 47∗ 47∗  22 22 22 22 22 22
V_{4} 15 36 1^{2} 1_{i} 47∗ 47∗ 22 22 22 22 22 22 22

Discriminant class: 1081

∞ 23 47 a_{1} a_{2} a_{3}  Over p = 23  Over p = 47  2 3 5 7 11 13 17 19
G h  a_{4}  f^{e} c_{w} d  s f^{e} c_{w} d  s 29 31 37 41 43 53 59 61
3. 2_{−} 3_{i} 1_{i}  1 9 82 1^{4} 3_{i} 23 23∗ 1^{2} 1_{i} 47∗ 47∗ 22 22 22 4 22 2 4 22
D_{4} 3  284  2 2 4 2 22 4 22 4